In (quantitative) genetics literature, heritability is usually defined (simplifying a bit) as the proportion of variance of a trait (lifespan, height, etc), in a population, that can be explained by genetics. The rest, by environmental factors, or error.
If height were a 100% heritability means that all differences in height between individuals would be explainable by genetics.
Maybe I should have been more explicit that here "variance explained by" is used quite literally with the meaning it has in statistical modeling. I get that you meant it in a "correlation not causation" way, but "correlation" in the context of statistics is a loaded term. Interpretability of heritability is difficult in general. Most people struggle with it. But if it truly were 100% (rarely is if ever) it would imply causation solely from genetic factors. Causation from both genetics and environment is always there - heritability just seeks to formalize the measure of proportion.
If height were a 100% heritability means that all differences in height between individuals would be explainable by genetics.